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arXiv:2208.11699

Decoding Measurement-Prepared Quantum Phases and Transitions: from Ising model to gauge theory, and beyond

Jong Yeon Lee, Wenjie Ji, Zhen Bi, Matthew P. A. Fisher

Published Aug 24, 2022• 7 citations
arXivHTMLPDF

Abstract

Measurements allow efficient preparation of interesting quantum many-body states with long-range entanglement, conditioned on additional transformations based on measurement outcomes. Here, we demonstrate that the so-called conformal quantum critical points (CQCP) can be obtained by performing general single-site measurements in an appropriate basis on the cluster states in $d\geq2$. The equal-time correlators of the said states are described by correlation functions of certain $d$-dimensional classical models at finite temperatures and feature spatial conformal invariance. This establishes an exact correspondence between the measurement-prepared critical states and conformal field theories of a range of critical spin models, including familiar Ising models and gauge theories. Furthermore, by mapping the long-range entanglement structure of measured quantum states into the correlations of the corresponding thermal spin model, we rigorously establish the stability condition of the long-range entanglement in the measurement-prepared quantum states deviating from the ideal setting. Most importantly, we describe protocols to decode the resulting quantum phases and transitions without post-selection, thus transferring the exponential measurement complexity to a polynomial classical computation. Therefore, our findings suggest a novel mechanism in which a quantum critical wavefunction emerges, providing new practical ways to study quantum phases and conformal quantum critical points.

DOI
10.48550/arxiv.2208.11699

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Measurements allow efficient preparation of interesting quantum many-body states with long-range entanglement, conditioned on additional transformations based on measurement outcomes. Here, we demonstrate that the so-called conformal quantum critical points (CQCP) can be obtained by performing general single-site measurements in an appropriate basis on the cluster states in d≥2d\geq2d≥2. The equal-time correlators of the said states are described by correlation functions of certain ddd-dimensional classical models at finite temperatures and feature spatial conformal invariance. This establishes an exact correspondence between the measurement-prepared critical states and conformal field theories of a range of critical spin models, including familiar Ising models and gauge theories. Furthermore, by mapping the long-range entanglement structure of measured quantum states into the correlations of the corresponding thermal spin model, we rigorously establish the stability condition of the long-range entanglement in the measurement-prepared quantum states deviating from the ideal setting. Most importantly, we describe protocols to decode the resulting quantum phases and transitions without post-selection, thus transferring the exponential measurement complexity to a polynomial classical computation. Therefore, our findings suggest a novel mechanism in which a quantum critical wavefunction emerges, providing new practical ways to study quantum phases and conformal quantum critical points.
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The impact of projective measurements on a many-body quantum state is tightly linked to its underlying entanglement structure. Critical states are particularly sensitive, as long-range entanglement allows local measurements to have global consequences. This has been extensively studied in the context of critical states described by a conformal field theory (CFT), where measurements can alter critical properties in post-measurement quantities ('measurement-altered criticality') and induce long-range entanglement with universal features ('measurement-induced entanglement'). By contrast, little is known about the role of measurements on critical states with quenched disorder, which admit no CFT description. Here, we develop a theory of measurements on one-dimensional critical states governed by infinite-randomness fixed points (IRFPs), focusing on two closely related examples: the random XXZ chain and the random transverse-field Ising model. We show that the measurement-induced entanglement decays as a power-law with universal exponent (3−5)/2(3-\sqrt{5})/2(3−5​)/2 in both models, and that the minimum number of measurements required to establish this entanglement obeys a universal scaling form. When only a finite density of measurements is made, we show that the critical properties of the state are remarkably robust: `measurement-altered criticality' is absent, with measurements either preserving the critical exponents, or destroying the criticality entirely. These results establish IRFPs as an analytically tractable arena for measurements on critical states, complementary to the CFT case. The outcome randomness that necessitates replica methods in clean systems becomes trivial at infinite randomness, with the universal response governed by the statistics induced by the quenched disorder.
Measurements can qualitatively alter correlations and entanglement emerging in gapless quantum matter. We show how a single round of measurements on gapless quantum systems can, upon rotating the measurement basis, induce nontrivial transitions separating regimes displaying universal characteristics governed by distinct boundary conformal field theories. We develop the theory of such “measurement-induced boundary transitions” by investigating a gapless parent of the one-dimensional cluster state, obtained by appropriately symmetrizing a commuting projector Hamiltonian for the latter. Projective measurements on the cluster state are known to convert the wave function, after post-selection or decoding, into a long-range-ordered Greenberger-Horne-Zeilinger (GHZ) state. Similar measurements applied to the gapless parent (i) generate long-range order coexisting with power-law correlations when post-selecting for uniform outcomes, and (ii) yield power-law correlations distinct from those in the pre-measurement state upon decoding. In the post-selection scenario, rotating the measurement basis preserves long-range order up until a critical tilt angle marking a measurement-induced boundary transition to a power-law-ordered regime. Such a transition—which does not exist in the descendant cluster state—establishes new connections between measurement effects on many-body states and nontrivial renormalization-group flows. We extend our analysis to tricritical Ising and three-state Potts critical theories, which also display measurement-induced boundary transitions, and propose general criteria for their existence in other settings.
Performing measurements on gapless quantum ground states can give rise to new universal phenomena, whose origins remain poorly understood even for the paradigmatic Ising critical point. Replica methods offer a natural framework to study such post-measurement ensembles, with the replica limit Q→1Q\to 1Q→1 describing physical Born-averaged observables. Here we show that, for the monitored critical Ising chain, the replicated post-measurement ensemble of states itself undergoes a phase transition precisely at the physically relevant replica limit Q→1Q \to 1Q→1, preventing standard replica treatments. We use boundary conformal field theory (BCFT) methods to explain the underlying mechanism of this transition. Our methods yield a perturbative analytic calculation of the universal O(1)O(1)O(1) constant in the Shannon entropy of the measurement record, previously known only numerically. We also apply our tools to study measurement-averaged spin correlators, providing a unified description of the universal properties of measurement outcomes and post-measurement states in the critical Ising chain.
Quantum information theory (QIT) is an emerging field of physics which aims to develop new methods of dealing with information by harnessing the power of quantum mechanics. Besides its potential to revolutionize the techniques of information processing and communication, it also provides novel approaches to better comprehend the foundations of quantum mechanics. Among many important problems in QIT, manipulation and dynamical characterization of correlations present in quantum systems stand out due to their relevance for the practical applications of the theory. This thesis intends to explore such correlations of quantum and classical nature from various perspectives. In particular, our discussions involve the investigation of local transformations among a class of entangled states and the examination of correlation measures in some physical models. We first examine the classification of the flip (0-1) and exchange symmetric (FES) states under local quantum operations. We study the optimal local one-shot conversions of FES states to determine the entanglement transformations that relate multiqubit FES states with the maximum possible probability of success. Next, we investigate the exchange symmetry properties of certain symmetric states when the qubits evolve according to a dephasing model which is also invariant under swap operation. We find that there exist states which do not preserve the exchange symmetry with unit probability during the time evolution, leading to the spontaneous breaking of exchange symmetry. Later, we turn our attention to the dynamics of quantum and classical correlations for qubit-qutrit systems in independent and global dephasing environments. In these cases, we demonstrate several interesting phenomena such as the transition from classical to quantum decoherence. Lastly, we investigate the thermal quantum and total correlations in the one-dimensional anisotropic XY model in transverse field. We discuss the ability of different measures to estimate the critical point of the quantum phase transition at finite temperature. We also consider the relation between correlations and the factorized ground state in this model. Furthermore, we study the effect of temperature on long-range correlations.
We explore the dynamical behavior at and near a special class of two-dimensional quantum critical points. Each is a conformal quantum critical point (CQCP), where in the scaling limit the equal-time correlators are those of a two-dimensional conformal field theory. The critical theories include the square-lattice quantum dimer model, the quantum Lifshitz theory, and a deformed toric code model. We show that under generic perturbation the latter flows toward the ordinary Lorentz-invariant (2+12+12+1)-dimensional Ising critical point, illustrating that CQCPs are generically unstable. We exploit a correspondence between the classical and quantum-dynamical behavior in such systems to perform an extensive numerical study of two lines of CQCPs in a quantum eight-vertex model or, equivalently, two coupled deformed toric codes. We find that the dynamical critical exponent zzz remains 2 along the U(1)-symmetric quantum Lifshitz line, while it continuously varies along the line with only Z2{\mathbb{Z}}_{2}Z2​ symmetry. This illustrates how two CQCPs can have very different dynamical properties, despite identical equal-time ground-state correlators. Our results equally apply to the dynamics of the corresponding purely classical models.